Dynamic analysis of cylindrical foundations under torsional loading via generic discrete-element models simulating soil stratum

Torsional vibration, considering soil-structure interaction, is essential to the dynamic response of most irregular structures. A systematic method is developed to seek the optimal simplified model among multiple model candidates for uniform soil on rigid base regarding dynamic soil-foundation interactions. A generic model is identified by the proposed method to simulate the cylindrical foundation resting on or embedded in the soil stratum under torsional vibrations. Various soil-foundation parameters, mainly including embedment depth, layer depth, and mass ratios, are considered in the simplified analysis. The frequency-magnification curves and resonant responses of the foundation using the generic model agree well with theoretical solutions. The resultant resonant magnification factors against mass ratios clearly illustrate the impacts of the whipping effect resulting from the soil-foundation interactions. The generic model performs better and adopts fewer parameters than the existing model to simulate the soil-foundation interactions. In addition, dimensionless parametric charts are presented to estimate foundation responses for engineering applications quickly. The proposed charts also significantly overcome the limitations of the Wolf and Paronesso model. The generic model shows efficiency and accuracy in simulating the soil stratum. This research could contribute to the foundation vibration analysis for torsional responses.


Developed systematic method
This section shows a systematic method to generate lumped-parameter models and to find the optimal model to simulate a cylindrical foundation on a uniform stratum undergoing harmonic torsional force.The soil stratum is viscoelastic and entirely connected to the cylindrical foundation overlaying or embedded in a homogenous soil layer on a rigid base, as shown in Fig. 1a,b.The cylindrical foundation that has a radius R is overlaying and embedded in a soil layer on rigid base, subjected to a harmonic torque M 0r .The target soil condition is considered to be a uniform stratum on a rigid base with a depth of H where the shear-wave velocity, V s , is uniformly distrib- uted; the Poisson ratio, ν, is 0.33; the damping ratio, ζ, is 0.05; G is the shear modulus; E is the embedded depth of a foundation.ρ is the soil mass density and V s is the shear-wave velocity of the soil.This research considers the SSI behavior using a foundation impedance function describing the force-displacement relation at the interface between the soil and the foundation.Note that the foundation represents one part of the upper structures.
For parametric analysis, a 0 = ωR/V s is the dimensionless frequency of harmonic torsional force with forcing frequency ω ; T is defined as embedment ratio (i.e., T = E/R); D is defined as layer depth ratio (i.e., D = H/R); b is the dimensionless mass ratio for torsional motion, (i.e., mass ratio in short in the following of the paper) and is non-dimensionalized as b = I r /ρR 5 , where I r is the polar mass moment of inertia of the foundation around the axis of rotation.It is noted that the polar mass moment of inertia is applied as a mechanism for quantifying the soil-foundation interaction.

Model candidates
The systematic method comprises seven simplified models, which are developed to simulate the uniform soil on a rigid base subjected to torsional vibration, as shown in Fig. 2. The simplified models are arranged sequentially from Model 1 to Model 7, as shown in Fig. 2a-g.Each of the seven simplified models connects to the foundation and the rigid base.The maximum internal degree of freedom is two for all simplified models where ψ 0 is the rotation of the foundation about the vertical axis of symmetry and ψ a is the additional rotational DOF about vertical axis.The simplified models are developed to simulate the relationship of the soil-foundation interaction behavior.The modeling elements of the proposed models include lumped masses moment of inertia, viscous dampers, and linear springs.The model parameters contain static torsional stiffness, K er , dynamic torsional stiffness, K d , the polar mass moment of inertia, M er , and torsional damping, C er .Each simplified model considers only three parameters: K er , C er , and either K d or M er .The layout of each proposed model is summarized as follows: Model 1 consists of three elements connected in parallel.Model 2 has three elements and one internal DOF contributed by a serial connection of a spring and a damper.Similarly, Model 3 also has three elements and one internal Table 1.Comparison of previous related research by the authors.

Research work Vibration mode Model candidates Model limitation for vibrating mass
Present study Torsional mode 7 None (Independent of mass ratios) Ref. 27 Horizontal and rocking modes 276 Depend on mass ratios Ref. 28 Vertical mode 2 None (Independent of mass ratios) Ref. 29 Vertical mode 33 Depend on mass ratios DOF given by a damper connected with a polar mass moment of inertia.Model 4 and Model 5 include four elements and one internal DOF.They are featuring in having two torsional dampers.Model 6 and Model 7 consist of four and five elements, respectively, while two polar masses moment of inertia are included in their layout.The parameters of the proposed models could be non-dimensionalized as follows: The systematic method calculates the parameters of simplified models using equivalent principles for static and dynamic responses.The adoption of equivalent theories is based on equaling the dynamic impedance of unbounded soils to the torsional impedance function of the simplified models; thus, it is assumed that the cylindrical foundation is massless.The torsional impedance of a target soil-foundation system is used in this study to calculate the model parameters for equivalent models.Based on the vibration theory, when each simplified model is subjected to a harmonic torque, the torsional impedance function of each model is defined as follows: where K sr is the static stiffness; k r and c r are the dynamic stiffness and damping coefficients, respectively.
(1)  www.nature.com/scientificreports/ The systematic method proposed uses three equivalent criteria to calculate the model parameters.Assume the torsional impedance function of a soil-foundation system is K r = K sr (k r + ia 0 c r ) .By equalizing the impedance function of a soil-foundation system to that of a simplified model, the equivalent equations regarding static stiffness, dynamic stiffness coefficients, and dynamic damping coefficients are established as follows: where K sr , k r , c r are the static stiffness, the dynamic stiffness coefficients, and the dynamic damping coefficients, respectively, for the target soil-foundation system subjected to torsional vibration.The first equivalent equation adopts the static equilibrium of a soil-foundation system.Therefore, the static stiffness K er of each model is derived by Eq. (3), i.e., Besides, Table 2 shows the impedance function of each simplified model, which is derived from dynamic equilibrium equations based on structural dynamics.A system of nonlinear equations (SNE) is established by substituting k r and c r of a model into Eqs.( 4) and (5).The other model parameters are further obtained by solv- ing the SNE in this study.

Frequency-response curves
This subsection aims to illustrate the frequency-response curve of a dynamic system.As the target soilfoundation system shown in Fig. 1 is subjected to a dynamic load, the equation of motion is expressed as follows: where M 0r is the harmonic torque, I r is the polar mass moment of inertia of the foundation around the axis of rotation, K r is the torsional impedance function of the soil-foundation system.Recall that the torsional impedance function is given by K r = K sr (k r + ia 0 c r ) .Equation ( 7) can be used to derive the steady-state responses of the foundation by assuming M 0r = m 0r exp(iωt) and ψ 0 (t) = φ 0 × exp(iωt) , as shown below.
where m 0r is the amplitude of the harmonic torque; M r and θ r are the dynamic magnification factor and the phase angle.
It is noted that the damped soil-foundation system considers the effect of both material damping and radiation damping on the foundation response.In detail, the effect of the material damping and radiation damping are respectively considered in the real and imaginary part of the impedance function for the target soil-foundation system, as shown in Fig. 1.The theoretical impedances by Tassoulas and Kausel 31 .are adopted for the surface foundation at the depth (D = 2) as the corresponding frequency a 0 = 0.1 to 6.2 with an interval of 0.1.On the other hand, the theoretical impedances by Tassoulas 32 are adopted for the foundation embedded at the depth (T = 1), (3) www.nature.com/scientificreports/layer depth ratios (D = 2, and 3) as the corresponding frequency a 0 = 0.1 to 3 with an interval of 0.1.Similarly, when using the proposed models (i.e., Model 1 to Model 7) to simulate the soil-foundation system, the dynamic magnification factor of the foundation could be written as follows: where the impedance coefficients k r and c r are given in Table 2. Therefore, the magnification factor of all simplified models can be calculated by Eq. ( 10) at each dimensionless frequency for a given mass ratio.

Optimal model by a single target approach (STA)
A single target approach (STA) is illustrated to find an optimal model for a target soil-foundation system regarding an individual mass ratio.At first, the parameters of the 7 model candidates are calculated at each dimensionless frequency.As there are NF dimensionless frequencies, the systematic method establishes 7 × NF simplified models.An error function ε r is defined as follows to search for the optimal equivalent model that reproduces the most accurate frequency-response curve.
where M r is the dynamic magnification factor of the target system given by Eq. ( 9); M r is the dynamic magnification factor calculated from the model parameters; P i is the weighting at the ith frequency point and Once the impedance functions of a soil-foundation system are given, the three model parameters for each model candidates are directly calculated from Eqs. ( 4)- (6), and the optimal model with the least value of error function is to be found by implementing Eq. (11).In brief, the procedure of STA considering single mass ratio is summarized as follows: 1. Find the torsional impedance function for a target soil-foundation system.2. Use Eq. ( 6) to compute the static stiffness of simplified models.
3. Use Eqs. ( 4) and (5) to solve the other model parameters.Subsequently, 7 × NF simplified models will be constructed for NF frequency points.4. Run an error analysis using Eq. ( 11) for each model and determine the optimal model with the minimum error. ( Error index of simplified models for surface cylindrical foundations.*T = E/R is an embedment ratio, D = H/R is a layer depth ratio, b is a mass ratio.N/A indicates no real number solution is found.Significant values are in bold.

Dimensionless parameters
Error index

A generic model for simulating the target soil-foundation system
This section pays attention to selecting the optimal equivalent model based on the procedure of STA among seven simplified models, which simulate the soil-foundation system in Fig. 1.According to the analyzed results for single mass ratio, Tables 3 and 4 summarizes the error index of all simplified models regarding various simulating conditions.A total of 63 analyzed cases are conducted with different embedment depth ratios (T = 0, and 1), layer depth ratios (D = 2, and 3), and mass ratios (b = 1, 5, and 10).It is observed in Table 3 for surface foundations that optimal simplified models are selected to investigate the target stratum.Model 3 could outperform almost the rest of the six model candidates for the analyzed cases as mass ratio b is smaller than 10.It is noted that in the case of mass ratio b equaling 10, a minimal discrepancy of error index is found between two model candidates (Model 3 and Model 1).On the other hand, it is observed clearly in Table 4 for the embedded cylindrical foundations that ) is the generic model to simulate the target sol-foundation system as the mass ratio of the foundation is smaller than 10.Furthermore, Model 3 shows relatively broader and more accurate adaptability than other model candidates in simulating the target soil-foundation system.Therefore, Model 3 is selected as a generic model in this study to simulate the dynamic interactions between the cylindrical foundation and the soil medium.All the following results are analyzed using Model 3.

Model validations on frequency-response curves
This section aims to verify the frequency-response curves calculated by the generic model and introduce related research considering cylindrical foundations overlying uniform soil on a rigid base.Figure 3 shows that the Wolf and Paronesso model 25 uses 3 degrees of freedom and eight parameters to simulate the SSI system (as shown in Fig. 1).The frequency-response curves and the peak response of the foundation by the generic model are compared and validated to those by the existing model.Wolf and Paronesso model 25 was applicable as the dimensionless frequency a 0 ≦ 4.1 for surface foundations and as a 0 ≦ 2.5 for embedded foundations.By adopting the theoretical impedance and least-error analysis, this research breaks through the frequency limitation of the Wolf and Paronesso model 25 .This work effectively extends the dimensionless frequency (a 0 ) from 4.1 to 6.2 for surface foundations and 2.5 to 3 for embedded foundations, as shown in Table .5.

Surface cylindrical foundations
The generic model uses the impedance function from Tassoulas and Kausel 31 (1983) to generate lump-parameter models for simulating the target stratum.Based on the calculation of the optimal equivalent model, numerous simplified models are generated using equivalent principles, and then the optimal equivalent model is obtained.
Figure 4 shows the relationship of frequency versus response for a cylindrical surface foundation (T = 0, D = 2).The generic model generates results consistent with the dynamic magnification factors obtained by the theoretical solutions in the cases where the mass ratio increases from 1, 5 to 10, which validates the accuracy of the generic model.It is particularly noted in Fig. 4a that as the mass ratio b is small (i.e., b = 1), the generic model simulates the dynamic magnification factor at 1 < a 0 < 4 more precisely than Wolf and Paronesso model 23 .Furthermore, Fig. 4 demonstrates that the generic model simulates more precisely than the existing model in the peak responses as the mass ratio varies from 1, 5 to 10.The reason for frequency-response differences might be that the curve-fitting method by Wolf and Paronesso model 25 used a different error weighting function to approach the theoretical impedance.In contrast, the generic model considers the effect of mass ratios to approach the frequency-response curve.

Embedded cylindrical foundations
The generic model uses the impedance function by Tassoulas 32 (1981) to construct simplified models for simulating target stratum.There are 30 given frequency points of a 0 .A series of simplified models are generated using equivalent principles; then, the optimal equivalent model is obtained by Eq. (11).   Figure 5 shows the trend of the frequency-response relationship for the cylindrical foundation (T = 1, D = 2).The generic model and the Wolf and Paronesso model 25 produce results consistent with the magnification factors calculated by the theoretical impedances.It is worth mentioning that the generic model adopting only three parameters performs more effectively than the existing model.
Figure 6 shows the trend of the frequency-response relationship for the cylindrical foundation (T = 1, D = 3).It is observed by compared with Fig. 5 as the layer depth ratio (D) increases, the generic model also produces results consistent with the magnification factors calculated by the theoretical impedances as the mass ratio varies from 1, 5 to 10, which validates the accuracy of the generic model.On the other hand, it is observed in Fig. 6 that the Wolf and Paronesso model 25 seems to underestimate the peak responses in all mass ratios slightly.

Model validations on resonant responses
This section focuses on validating the resonant responses obtained by the generic model (i.e., Model 3), which is significant for engineering applications to avoid the coincidence of the forcing frequency with the natural frequency of the soil-foundation system.The resonant response of the foundation consists of the maximum magnification factor and the corresponding frequency.The discontinuity of the adopted impedance function shows discrete characteristics 18 ; therefore, catching the peak response of the foundation between discrete frequency points appears necessary and significant.The five-point interpolation technique 30 is adopted to catch the peak responses of the theoretical solution by Tassoulas and Kausel 31 (1983) and Tassoulas 32 (1981).In addition, as Wolf and Paronesso model 25 (1992) calculates the dynamic response by evaluating the torsional impedance functions based on its model parameters, the five-point interpolation approach 30 is also applied to Wolf and Paronesso model 25 to calculate the peak displacements.However, the peak displacements using the generic model do not perform the five-point interpolation approach 30 because the three model parameters vary with the mass ratio.To catch the approximate peak response for the foundation, this study adopts the impedance function with expanded frequencies (i.e., the interval of a 0 is 0.1) by a standard third-order interpolation function.Thus, the analyzed peak reactions of the foundation can be directly selected among the calculated discrete responses.www.nature.com/scientificreports/

Surface cylindrical foundations
Figure 7a (T = 0, D = 2) shows that the resonant frequencies of the torsional foundation response calculated by both the generic model and the Wolf and Paronesso model 25 are consistent well with the theoretical solutions as the torsional mass ratios of the foundation are in the range (b = 2 to 10).On the other hand, it is comparatively observed that the generic model and the Wolf and Paronesso model 25 could underestimate the resonant frequencies as the torsional mass ratios of the foundation are small (b = 0 to 2).In contrast, the generic model evaluates the resonant frequencies more accurately than the existing model.Figure 7b (T = 0, D = 2) shows that resonant magnification factors of the torsional foundation response calculated by the generic model are consistent well with the theoretical solutions as the torsional mass ratio of the foundation varies in the range from 0 to 10.In addition, it is comparatively observed that the generic model simulates more precisely.At the same time, the Wolf and Paronesso model 25 significantly underestimates the resonant magnification factors of the torsional foundation responses as the torsional mass ratios of the foundation are larger (b = 3 to 10).The reasons that caused these differences in the resonant magnification factor might be that the peak magnification factor is fairly sensitive to large mass ratios, especially as the layer depth ratio of the soil stratum is small, and that the generic model adopts the optimal equivalent model which carefully considers the effect of each single mass ratio on the peak response.
Note that Fig. 7(b) (T = 0, D = 2) shows the results for a relatively small layer depth ratio of the soil stratum.A remark is that the analyzed results of the resonant magnification factors simulated by the generic model show agreement with the theoretical solutions and are consistent with the whipping effect of the soil-structure interaction 33 .the Wolf and Paronesso model 25 could respectively overestimate and underestimate the resonant frequencies.

Embedded cylindrical foundation
Whereas the Wolf and Paronesso model 23 evaluates the resonant frequencies slightly more accurately than the generic model as mass ratios are small (b = 0 to 4). Figure 8b (T = 1, D = 3) shows that as the torsional mass ratios of the foundation are in the larger range (b = 4 to 10), resonant frequencies of the torsional foundation response calculated by the generic model are consistent well with the theoretical solutions while the Wolf and Paronesso model 25 slightly overestimates.However, as the torsional mass ratios of the foundation are in the smaller range (b = 0 to 4), the generic model slightly overestimates the resonant frequencies while the Wolf and Paronesso model 25 slightly underestimates resonant frequencies.Generally, the generic model and the Wolf and Paronesso model 25 simulate the resonant frequencies in a way that approaches theoretical solutions as the mass ratio increases.Figure 9 (T = 1, D = 2 and 3) shows that resonant magnification factors of the torsional foundation response simulated by the generic model and the Wolf and Paronesso model are both consistent well with the theoretical solutions as the torsional mass ratio of the foundation varies in the range from 0 to 10.In addition, it is comparatively observed in Fig. 9b that the generic model outperforms the existing model in terms of the Wolf and Paronesso model 25 slightly underestimating the resonant magnification factors of the torsional foundation at all mass ratios considered (b = 0 to 10).The generic model performs better in the resonant magnification factor because the resonant magnification factor of the theoretical solution is precisely detected by the five-point interpolation method.Moreover, the generic model considers the effect of mass ratios on the dynamic magnification factor.The generic model seems to estimate better the resonant magnification factor for surface and embedded foundations.

Multiple target approach (MTA) for practical applications
The STA illustrated in the previous section is limited to the application for a single mass ratio.In light of STA's limitations, a multiple target approach (MTA) is developed, as listed in Supplementary Information-SCP, to enhance the engineering application of the proposed model.This section aims to provide reference charts for engineering use to build the generic model quickly.The Wolf and Paronesso model 25 was developed in the 1990s.
It was a pioneering study at that time, but the model element was complex, and there was no relevant parametric chart for engineering applications.Therefore, this study further proposes the MTA to find a generic model and consider a broader range of soil-foundation parameters to build related parametric charts, making the simplified model easier to apply in engineering.

Dimensionless parametric charts
In this subsection, parametric charts are established based on the procedure of MTA listed in Supplementary Information-SCP, considering multiple mass ratios.Consider that the maximum dimensionless frequency is 3 with a discrete interval of 0.1, then the number of dimensionless frequency (NF) is 30.Assume that a group of mass ratios (b = 0.5,1,2,3,4,5) is assigned, then the number of multiple mass ratios (NB) is 6.Applying Supplementary Information-SCP, 7 × NB × NF (i.e., 7 × 6 × 30) sets of simplified models are generated, and the model parameters of the optimal equivalent model are obtained at each combination of the embedment depth ratio (T), and the layer depth ratio (D).The embedment depth ratios are T = 0, 0.25, 0.5, 1, 1.5, and 2. The minimum layer depth ratio is defined as D min = T + 0.25, and the maximum layer depth is defined as D max = 10 with an integral interval of 1 from the minimum.After analyzing the optimal equivalent model for multiple mass ratios considering various soil and foundation parameters, the analyzed results determine that the optimal equivalent model is solely the model candidate (Model 3).The result that Model 3 is the optimal simplified model among the 7 model candidates is consistent with that in Tables 3 and 4. Recall that Model 3 is selected as the generic model for generating corresponding dimensionless charts.In addition, Table 5 shows that the proposed model in this research considers a broader range of main soil-foundation parameters than the Wolf and Paronesso model 25 (1992).Furthermore, Fig. 10a-c are generated as dimensionless charts of the static stiffness coefficient ( k er ), damping coefficient ( c er ), and mass coefficient ( m er ), respectively, for the genetic model.Those referenced values in the chars correspond to the referenced scenarios to compute the dynamic responses of foundations.Error analysis of applying the charts will be substantially introduced in the next subsection, followed by an illustrative example for practical engineering application.

Error analysis and applicable scope
This subsection compares the errors by different approaches and gives suggestions for application.The STA follows the process described in the previous section to build the optimal model for a single mass ratio.The MTA applies the dimensionless parametric charts given in Fig. 10 to create the optimal model quickly.The optimal models generated by STA and MTA may have different performances in simulating the target system.Thus, it is essential to investigate the difference between the two approaches.Eqs. ( 12) and ( 13) define a relative error index for the STA, ε STA and for the MTA, ε MTA , respectively.
where M system i,j , M STA i,j , and M MTA i,j indicate the magnification factors calculated respectively using SASSI 30 program, the STA, and the MTA for the ith mass ratio and jth dimensionless frequency ratio.
The assumption of multiple mass ratio combination is b = 0.5, 1, 2, 3, 4, and 5. Thus, based on the maximum and minimum of multiple mass ratios, the comparison of error index that corresponds to mass ratio b = 0.5, 1, 2.5, and 5 are selected.It is noted that each relative error index corresponds to each combination of the embedment depth ratio (T), the layer depth ratio (D), and the mass ratio (b).
Figure 11 shows the error index, by Eqs. ( 12) and ( 13), of surface foundation considering various soilfoundation conditions (T = 0, D = 0.5-10, b = 0.5, 1, 2.5, and 5).Remarks are made as follows: • The STA shows a good agreement with the SASSI 30 solution and a precision higher than 91.1%, as the maximum error occurs at T = 0, D = 1, and b = 2.5.• The MTA shows a precision of 42.1% ~ 97.1% with the numerical solutions, as the maximum error occurs at T = 0, D = 0.5, and b = 0.5.However, adjusting the application range to 4≦D≦10 and 1≦b≦5 is suggested to demonstrate a precision higher than 90%.

Illustrative example
Consider a rigid cylindrical foundation embedded in uniform soil on a rigid base, as shown in Fig. 1.The generic model (i.e., Model 3 shown in Fig. 2) is developed herein to simulate the interactions between the foundation and the soil medium.The properties of the target system are described as follows: • Foundation geometry: a radius of the foundation, R = 2.5 m, and an embedment depth, E = 2.5 m.For the target system considered, the embedment ratio T = 2.5/2.5 = 1 and the layer depth ratio D = E/R = 12.5/2.5 = 5.Subsequently, the dimensionless coefficients k er , c er , and m er are directly obtained by que- rying Fig. 10, i.e. k er = 19.83,c er = 10.68, and m er = 8.94.Hence, the model parameters for the soil-foundation system can be evaluated through Eq. (1), as shown below.www.nature.com/scientificreports/ The process above may show the effectiveness of the parametric charts developed for engineering applications.

Conclusion
A systematic method is developed to investigate lump-parameter models for various homogeneous soil strata on rigid base, considering the dynamic interaction behavior of soil-foundation systems.The systematic method proposes seven simplified models comprising springs, dampers, and lumped mass.This research utilizes the proposed method to adaptively identify a generic model to investigate the dynamic response of a cylindrical foundation embedded in uniform soil on rigid base under torsional vibrations.The generic model simulates the foundation response considering embedment depth ratios, layer depth ratios, and mass ratios.The validation of the generic model to simulate the dynamic frequency-magnification curve and resonant behavior has been confirmed.The generic model effectively overcomes the limitation of the Wolf and Paronesso model about the upper-frequency limit in frequency-response curves.In addition, the existing model underestimates the resonant magnification factors when the foundation mass ratio increases.
Moreover, the proposed model has broader applicability than the Wolf and Paronesso model and has also achieved a precision higher than 90% within the suggested application range.Dimensionless parametric charts are proposed to estimate the model parameters readily and to efficiently build the generic model in the time domain for the dynamic analysis of cylindrical foundations under torsional load.The research results showed the limitation of the existing model may be improved, as follows: the foundation embedment deeper than the foundation radius (R), the depth of the soil layer greater than 3R, and the dimensionless frequency ratio larger than 2.5.
This research developed a generic simplified model with better practical applications than the Wolf and Paronesso model.Additionally, this study provides the single target approach (STA) and the multiple target approach (MTA) to find the optimal equivalent models for simulating target soil-foundation systems.The STA establishes an optimal model for a target system vibrated with a fixed mass.The MTA shows excellent potential in engineering applications regarding various soil-foundation conditions.The generic model may be applied efficiently and accurately to simulate the dynamic torsional responses for the cylindrical foundation embedded in uniform soils on rigid base.For more complicated layers of soils, additional discrete-element modelings may be further extended to enhance the capabilities of the generative models.

Figure 1 .
Figure 1.Soil-foundation system subjected to torsional vibration.(a) cylindrical resting on uniform soil on rigid base, (b) cylindrical embedded in uniform soil on rigid base.

Figure 2 .
Figure 2. Proposed simplified models with seven candidates.
Model name Embedded depth ratio Layer depth ratio Maximum dimensionless frequencyWolf and Paronesso model 25 0

Figure
Figure8a(T = 1, D = 2) shows that resonant frequencies of the torsional foundation response calculated by both the generic model and the Wolf and Paronesso model25 are consistent well with the theoretical solutions as the torsional mass ratios of the foundation are in the range (b = 4 to 10).On the other hand, as the torsional mass ratios of the foundation are in the range (b = 0 to 4), it is comparatively observed that the generic model and

Table 2 .
Dynamic impedance coefficients of the proposed simplified models.

Table 4 .
Error index of simplified models for embedded cylindrical foundations.one unique model candidate (Model 3) is consistently calculated to investigate the target stratum, it outperforms the other six model candidates for the analyzed cases of mass ratio, b, is smaller than and equal to 10.Based on the numerical analysis of the error index, a conclusion is made that Model candidate (Model 3 *T = E/R is an embedment ratio, D = H/R is a layer depth ratio, b is a mass ratio.N/A indicatesno real number solution is found.Significant values are in bold.Vol.:(0123456789) Scientific Reports | (2023) 13:19163 | https://doi.org/10.1038/s41598-023-46046-7www.nature.com/scientificreports/

Table 5 .
Comparisons of simulation parameters between the existing and proposed models.